Pramana – J. Phys. (2018) 90:71 © Indian Academy of Sciences https://doi.org/10.1007/s12043-018-1568-3 Multiple periodic-soliton solutions of the (3 + 1)-dimensional generalised shallow water equation YE-ZHOU LI1 and JIAN-GUO LIU1,2 ,∗ 1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China 2College of Computer, Jiangxi University of Traditional Chinese Medicine, Nanchang 330004, Jiangxi, China ∗Corresponding author. E-mail: [email protected] MS received 8 November 2017; revised 9 December 2017; accepted 27 December 2017; published online 30 April 2018 Abstract. Based on the extended variable-coefficient homogeneous balance method and two new ansätz functions, we construct auto-Bäcklund transformation and multiple periodic-soliton solutions of (3 + 1)-dimensional generalised shallow water equations. Completely new periodic-soliton solutions including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave are obtained. In addition, cross-kink three-soliton and cross-kink four-soliton solutions are derived. Furthermore, propagation characteristics and interactions of the obtained solutions are discussed and illustrated in figures. Keywords. Interactions; periodic-soliton solutions; (3 + 1)-dimensional generalised shallow water equation. PACS Nos 02.70.Wz; 05.45.Yv; 02.30.Jr 1. Introduction flows, tsunami prediction and so on, which was studied in different ways. By using the Nonlinear partial differential equations play important generalised tanh algorithm method with symbolic com- roles in nonlinear science [1–8]. They are widely used putation, Tian and Gao [27] obtained the soliton- to describe the phenomena in various fields such as type solutions of eq. (1). Zayed [28] presented the optical fibre communications, ocean engineering, fluid travelling wave solutions of eq. (1)bythe(G/G)- dynamics, plasma physics, chemical physics, etc. The expansion method. Tang et al [29] derived the Gram- past work is mainly concerned with the solutions [9– mian and Pfaffian derivative formulae, the Grammian 13]. A variety of analytical and numerical methods have and Pfaffian solutions of eq. (1) by the Hirota bilinear been proposed for studying soliton models, including form. inverse scattering method [14], Bäcklund transforma- In this work, we shall introduce new ansätz functions tion [15], Hirota direct method [16], homogeneous to study the (3 + 1)-dimensional generalised shallow balance method [17–19], F-expansion method [20], water equations and discuss their multiple periodical- similarity transformation [21], etc. soliton solutions. Shallow water equations, also called Saint-Venant This paper is organised as follows: In §2,weper- equations in its unidimensional form after Adhémar form the extended variable-coefficient homogeneous Jean Claude Barré de Saint-Venant), are a category of balance method (EvcHB) on the (3 + 1)-dimensional hyperbolic partial differential equations (or parabolic if generalised shallow water equation and present a new viscous shear is considered) describing the flow below a auto-Bäcklund transformation of eq. (1). In §3,the pressure surface in a fluid (sometimes, but not necessar- non-travelling wave soliton-type solutions of the (3+1)- ily, a free surface) [22–24]. In this paper, we shall study dimensional generalised shallow water are investigated. the following (3 + 1)-dimensional generalised shallow In §4, the travelling wave multisoliton solutions are water equation [25,26]: obtained by using the three-wave approach, which con- tain many singular periodic soliton solutions, periodic u − u − 3u u − 3u u + u = 0, (1) yt xz x xy y xx xxxy cross-kink waves solutions, two-solitary solutions and where u = u(x, y, z, t). Equation (1) has applications doubly periodic solitary solution. Finally, conclusions in weather simulations, tidal waves, river and irrigation are given in §5. 71 Page 2 of 11 Pramana – J. Phys. (2018) 90:71 2. Auto-Bäcklund transformation −6u ξ 3 + 3ξ ξσ(η)ξ 2 − 2ξ 2u 0y x y xx x 0x + 3 [5ξσ(η)− 2] ξ ξ In terms of the idea of EvcHB method [30], the solutions xy xx +[ξσ(η)+ 2]ξ ξ ξ = 0, (9) of eq. (1) can be written as follows: x xxy x − ξ 2 + ξ ξ − ξ + ξ + ξ ξ u(x, y, z, t)=ψ(y, z, t)∂xf[ξ(x, y, z, t), η(x, y, z, t)] 3u0xy x yt x 2 xz 6u0x xy 9u0y xx x + 4ξ ξ − ξ ξ + ξ (ξ − 2ξ ) + u0(x, y, z, t). (2) xxxy x z xx xy t xxx 3 In the following analysis, we shall stay with the follow- − ξσ(η)[ξσ(η)− 4] ξxxξxxy + ξxyξxxx ing conditions: 2 + ξy (ξxt − 3u0x ξxx − 3ξx u0xx + ξxxxx) = 0, (10) ψ(y, z, t) = 1,η(x, y, z, t) = η (x, y, z, t) = 0 x y ξxyt − 3u0xyξxx − 3ξxyu0xx − ξxxz ⇒ η(x, y, z, t) = η(z, t). (3) − 3u0x ξxxy − 3u0yξxxx + ξxxxxy = 0, (11) Substituting (2)and(3) into eq. (1), we obtain σ (η)(η ξ − η ξ ) = 0, (12) t xy z xx ξ ξ 4 + ξ ξ 3 − η + ξ ξ 2 u − 3u u − 3u u + u − u = 0. fξξξξξ y x 4 fξξξξ xy x z fξξη fξξξ z x 0xxxy 0x 0xy 0y 0xx 0yt 0xz 2 2 (13) + 6 fξξξξξ ξ ξ + 6 fξξξξ ξ y xx x xxy x + ηt fξξη + fξξξξt ξyξx + fξξξytξx From (2)and(6), the new auto-BT for the (3 + 1)- dimensional generalised shallow water equation can be − 2 fξξξxzξx + 12 fξξξξxyξxxξx written as + 4 fξξξξyξxxxξx + 4 fξξξxxxyξx + ξ ξ 2 + + ξ ξ − u(x, y, z, t) = [−2ln(ξ) + δ(η) + ξσ(η)] 3 fξξξ y xx u0yt fξξ y xt u0xz x + (η fξη + fξξξ )ξ + fξ ξ + u0(x, y, z, t), (14) t t xy xyt − η + ξ ξ − ξ + ξ ξ z fξη fξξ z xx fξ xxz 6 fξξ xx xxy with σ(η), δ(η) and ξ satisfying eqs (8)–(13). The + u + fξξξ ξ + fξ ξ meaning of auto-BT consisted of (8)–(14)isthatif 0xxxy y xxxx xxxxy 2 2 u0(x, y, z, t) is a special solution of eq. (1), then the − 3 fξξξ + u + fξ ξ fξξξξ ξ + u x 0x xx y x 0xy expression (14) is another solution of eq. (1). + fξξ(2ξ ξ + ξ ξ ) + fξ ξ x xy y xx xxy + 4 fξξξ ξ − 3 u + fξξξ ξ + fξ ξ xy xxx 0y y x xy × ξ 3 + ξ ξ + + ξ = . 3. Non-travelling wave soliton-type solutions fξξξ x 3 fξξ xx x u0xx fξ xxx 0 (4) ξ 4 Setting the coefficient of x in (4) to zero yields an ordi- Now we use the new auto-BT consisted of eqs (8)–(14) nary differentiable equation (ODE) for f ;thatis to obtain exact solutions of eq. (1). Starting from eq. (8), we obtain fξξξξξ − 6 fξξ fξξξ ξy = 0, (5) which admits the solution σ(η) = 0. (15) f =−2ln(ξ) + δ(η) + ξσ(η), (6) Aiming at the non-travelling wave soliton-type solu- = where δ(η) and σ(η)are differential functions. Accord- tions, we substitute u0 0 and a trial solution ing to (6), we obtain ( , ) + ( , , ) ξ(x, y, z, t) = 1 + em z t x n y z t (16) − ξσ(η) 2 2 [2 ] 2 1 fξ = fξξ, fξξ = fξξξξ, into eqs (9)–(13), where m(z, t) and n(y, z, t) are dif- 2 3 2 − ξσ(η) ferentiable functions. Substituting (15)and(16)into fξ fξξ = fξξξ, (9)–(13), we get 2 2 − ξσ(η) my fξ fξξξ = fξξξξ. (7) n(y, z, t) = C1(z, t) + , 3 C3(z) Substituting (5)and(7)into(4) yields a linear poly- mt (z, t) = C3(z)mz, nomial of fξ , fξξ,.... Equating the coefficients of fξ , m3 + C C (z, t) = 1t , (17) fξξ,...to zero, holds 1z ( ) C3 z ξσ(η)ξ 2 ξ ξ + ξ ξ = , x x xy y xx 0 (8) where C1(z, t) is an integrable function and C3(z) is an − ξ ξ 2 + ξ ξ ξ + ξσ(η)+ ξ ξ ξ 2 z x 2 t y x [3 2] y xxx x arbitrary differentiable function. Pramana – J. Phys. (2018) 90:71 Page 3 of 11 71 Collecting all the above terms, we can derive the Therefore, we obtain the following periodic general solutions of eq. (1) as follows: breather-type of kink three-soliton solutions for eq. (1) as follows: xm+ ymzm +C mt 1 ( , , , ) =−2me . γ u x y z t ym m (18) 2 1 xm+ z +C u1 = v0 − 2 k4α4 cosh xα4 − t α − α4 + σ4 1 + e mt 1 4 β 1 γ All parameters have been explained before. m = m(z, t) 2 1 − k2α2 sin xα2 + t α + α2 + σ2 and n = n(y, z, t) satisfy constraint (17). Solution (18) 2 β 1 contains more arbitrary parameters than the solution yβ1+zγ1+σ1 −yβ1−zγ1−σ1 obtained before in refs [27–29]. / e k1 + e γ + α + α2 + 1 α + σ k2 cos x 2 t 2 2 2 β1 4. Travelling wave multisoliton solutions zγ β + k yβ + 1 3 + σ 3 cosh 3 β 3 Aiming at the travelling wave multisoliton solutions, 1 we now suppose that σ(η) = 0, u = v and the real γ 0 0 + α − α2 − 1 α + σ . function ξ(x, y, z, t) has the following ansätz: k4 sinh x 4 t 4 4 4 (22) β1 θ1 −θ1 ξ(x, y, z, t) = k1e + e + k2 cos(θ2) The expression (22) is the periodic breather-type of kink + k3 cosh(θ3) + k4 sinh(θ4) , (19) three-soliton solutions of eq. (1) which is a periodic wave in x, y and meanwhile is a two-soliton in x − t − where θi = αi x + βi y + γi z + δi t + σi , i = 1, 2, 3, 4 and in y t, respectively.